Relatively Strictly Singular Perturbations, Essential Spectra, and Application to Transport Operators

AbstractThe stability of essential spectra of a closed, densely defined linear operator A on Lp-spaces, 1≤p≤∞, when A is subjected to a perturbation by a bounded strictly singular operator was discussed in a previous paper by K. Latrach and A. Jeribi (1998, J. Math. Anal. Appl.225, 461–485). In the present paper we prove the invariance of the Gustafson–Weidmann, Wolf, Schechter, and Browder essential spectra of A under relatively strictly singular (not necessarily bounded) perturbations on these spaces. Further, a precise characterization of the Schechter essential spectrum is given. We show that these results are also valid on C(Ξ) where Ξ is a compact Hausdorff space. The results are applied to the one-dimensional transport equations with anisotropic scattering and abstract boundary conditions